Georg Kantor: Set theory, biography and family mathematics

George Cantor (pictured below in the article) is a German mathematician who created the theory of sets and introduced the concept of transfinite numbers, infinitely large, but differing from each other. He also gave a definition of ordinal and cardinal numbers and created their arithmetic.

George Cantor: brief biography

He was born in St. Petersburg on 03/03/1845. His father was the Dane of the Protestant religion Georg-Valdemar Kantor, who was engaged in trade, including on the stock exchange. His mother Maria Boehm was a Catholic and came from a family of outstanding musicians. When, in 1856, Georg's father fell ill, the family moved to Wiesbaden, and then to Frankfurt, in search of a milder climate. The boy's mathematical talents appeared before his 15th birthday while studying in private schools and gymnasiums in Darmstadt and Wiesbaden. In the end, Georg Kantor persuaded his father in his firm intention to become a mathematician, not an engineer.

After a short study at the University of Zurich in 1863, Kantor was transferred to the University of Berlin to study physics, philosophy and mathematics. There he was taught:

• Karl Theodor Weierstrass, whose specialization in the analysis probably had the greatest influence on Georg;
• Ernst Eduard Kummer, who taught the higher arithmetic;
• Leopold Kronecker, a specialist in number theory, who subsequently opposed Cantor.

After one semester at the University of Göttingen in 1866, the next year Georg wrote a doctoral dissertation entitled "In Mathematics, the art of asking questions more valuable than solving problems" concerning the problem that Karl Friedrich Gauss left unresolved in his Disquisitiones Arithmeticae (1801) . After a brief teaching at the Berlin School for Girls, Kantor began working at the University of Halle, where he remained for the rest of his life first as a teacher, from 1872 as an assistant professor and from 1879 as a professor.

Research

At the beginning of a series of 10 works from 1869 to 1873, George Cantor considered the theory of numbers. The work reflected the enthusiasm for the subject, his studies of Gauss and the influence of Kronecker. At the suggestion of Henry Edouard Heine, Cantor's colleague in Halle, who recognized his mathematical talent, he turned to the theory of trigonometric series in which he expanded the notion of real numbers.

Starting from work on the function of the complex variable of the German mathematician Bernhard Riemann in 1854, in 1870 Cantor showed that such a function can be represented only in one way - by trigonometric series. Considering the totality of numbers (points) that would not contradict such a representation, led him, first, in 1872 to the definition of irrational numbers in terms of convergent sequences of rational numbers (fractions of integers) and further to the beginning of work on the work of his entire life, The theory of sets and the concept of transfinite numbers.

Theory of sets

George Cantor, whose set theory was born in correspondence with the mathematician of the technical institute of Braunschweig, Richard Dedekind, was friends with him from childhood. They came to the conclusion that sets, finite or infinite, are a collection of elements (for example, numbers, {0, ± 1, ± 2 ....}), Which possess a certain property while retaining their individuality. But when George Cantor applied a one-to-one correspondence to study their characteristics (for example, {A, B, C} to {1, 2, 3}), he quickly realized that they differed in their degree of belonging, even if they were infinite sets , That is, sets whose part or subset includes as many objects as it itself. His method soon produced surprising results.

In 1873, George Cantor (mathematician) showed that rational numbers, although infinite, are countable, because they can be put in a one-to-one correspondence with natural ones (i.e., 1, 2, 3, etc.). He showed that the set of real numbers, consisting of irrational and rational, is infinite and uncountable. More paradoxically, Cantor proved that the set of all algebraic numbers contains as many elements as the set of all integers, and that transcendental numbers that are not algebraic, which represent a subset of irrational numbers, are uncountable and, therefore, their number is greater than integers , And should be considered as infinite.

Opponents and supporters

But Cantor's work, in which he first put forward these results, was not published in the journal Krell, since one of the reviewers, Kronecker, was categorically opposed. But after Dedekind's intervention, it was published in 1874 under the title "On the characteristic properties of all real algebraic numbers."

Science and personal life

In the same year, during a honeymoon with his wife Valli Gutman in Interlaken, Switzerland, Cantor met Dedekind, who kindly responded to his new theory. George's salary was small, but with the money of his father, who died in 1863, he built a house for his wife and five children. Many of his works were published in Sweden in the new journal Acta Mathematica, whose editor and founder was Gesta Mittag-Leffler, among the first to recognize the talent of the German mathematician.

Relationship with metaphysics

Cantor's theory has become a completely new subject of research relating to the mathematics of infinite (for example, series 1, 2, 3, etc., and more complex sets), which largely depended on a one-to-one correspondence. Kantor's development of new methods of raising questions about continuity and infinity has given his studies an ambiguous character.

When he claimed that infinite numbers really exist, he turned to ancient and medieval philosophy with respect to actual and potential infinity, as well as to the early religious education that his parents gave him. In 1883, in his book The Foundations of General Set Theory, Cantor combined his concept with Plato's metaphysics.

Kronecker, who claimed that only whole numbers exist ("God created whole numbers, the rest is the work of man"), for many years, vehemently rejected his reasoning and prevented his appointment at the University of Berlin.

Transfinite numbers

In the years 1895-97. Georg Cantor fully formed his idea of continuity and infinity, including infinite ordinal and cardinal numbers, in his most famous work, published under the title "Contribution to the Creation of the Theory of Transfinite Numbers" (1915). This work contains his concept, to which he was led by the demonstration that an infinite set can be put in a one-to-one correspondence with one of its subsets.

Under the smallest transfinite cardinal number, he meant the power of any set that can be put in a one-to-one correspondence with natural numbers. Cantor called it alef-zero. Large transfinite sets are denoted alef-one, alef-two, etc. He further developed the arithmetic of transfinite numbers, which was analogous to finite arithmetic. Thus, he enriched the notion of infinity.

The opposition he faced, and the time it took for his ideas to be fully accepted, are explained by the difficulties of reevaluating the ancient question of what the number is. Cantor showed that the set of points on the line has a higher power than alef-zero. This led to the well-known problem of the continuum hypothesis - there are no cardinal numbers between the alef-zero and the power of the points on the line. This problem in the first and second half of the 20th century aroused great interest and was studied by many mathematicians, including Kurt Gödel and Paul Cohen.

Depression

George Cantor's biography from 1884 was overshadowed by the beginning of his mental illness, but he continued to work actively. In 1897, he helped to conduct the first international mathematical congress in Zurich. Partly because he was opposed by Kronecker, he often sympathized with young beginning mathematicians and sought to find a way to save them from harassment by teachers who feel threatened by new ideas.

Confession

At the turn of the century his work was fully recognized as the basis for the theory of functions, analysis and topology. In addition, the books of Cantor George served as an impetus for the further development of intuitional and formalistic schools of the logical foundations of mathematics. This significantly changed the teaching system and is often associated with "new mathematics".

In 1911, Kantor was among the invited to celebrate the 500th anniversary of St. Andrews University in Scotland. He went there hoping to meet with Bertrand Russell, who in his recently published work Principia Mathematica repeatedly referred to the German mathematician, but this did not happen. The University awarded Cantor the honorary degree, but due to illness he was unable to accept the award personally.

Cantor retired in 1913, lived in poverty and during the First World War starved. Celebrations in honor of his 70th birthday in 1915 were canceled because of the war, but a small ceremony took place at his home. He died on 06.01.1918 in Halle, in a psychiatric hospital, where he spent the last years of his life.

George Cantor: biography. A family

On August 9, 1874, the German mathematician married Vali Gutman. The spouses had 4 sons and 2 daughters. The last child was born in 1886 in the new house bought by Cantor. To support his family helped him to inherit his father. On the health of Cantor greatly affected the death of his youngest son in 1899 - since then he was not depressed.